 Hello everyone. Welcome to yet another session of our NPTEL on non-linear and adaptive control. I am Srikanth Sukumar from Systems and Control, IIT Bombay. We are of course well into our third week and as we saw last time, we have covered several important notions which include stability, uniform stability and attractivity properties. And we are of course going to be starting to be able to look at very, very special properties, that is property of stability and attractivity for dynamical systems in close loop for algorithms that are driving systems such as what you see in my background. So, without delaying any further, let us look at the specifics of what we actually learned last time. So, last time, we actually started with the van der Paul oscillator and we just used the face plane plots like you can see here with the x here and x dot here or x and y here, whatever you want to name the states as. So, the face plane plot showed to us that all trajectories starting at origin start to head towards the limit cycle. And similarly, all trajectories starting outside also start to head towards the limit cycle for certain values of mu. That of course we did not see in any detail. That is not of interest right now. So, rather interesting and we saw that it does not matter what the value of mu is, all it does is it changes the shape of this limit cycle. But eventually, your origin still remains unstable. So, which was rather interesting for us. Now, after that we since we had already seen the ideas of stability and uniform stability, we wanted to go to the other set of properties which relate to somehow system convergence. Which is the sort of property that we had verified using the Babalat's lemma on the spring mass damper system a few weeks ago. So, these properties are of course were of course named attractivity in the context of systems theory. And we defined three different attractivity properties. For the first one was called attractivity, the second uniform attractivity and the third was global uniform attractivity. Now, we also pointed out of course that there was no such thing as global stability. Because we have stable and uniformly stable, there is no notion of a global stability. But with attractivity, we also have a notion of global attractivity. We also verified each of these for the Masera example that is this particular example that we were working out. We sort of verified which property it satisfies. And we saw that it is at best globally asymptotically stable. So, of course after seeing attractivity and global attractivity and global uniform attractivity and so on. We also spoke of the most desirable set of properties which are the ones that are here. So, all these are in fact the most desirable properties. The first one and why do we come to these at the end is because these are simply combination of the already you know, proven properties of stability and attractivity. And therefore, we don't need to define each of them in epsilon delta terminology because we've already done that for attractivity and stability. So, if you see the first one was asymptotic stability and I use the acronyms rather frequently. So, I actually wrote them out for you at the left side, left the left hand side of this. So asymptotically stable or AS requires stability plus attractivity. Then we had I actually put in something in the middle here because the Masera system was of course more than asymptotic stable. So therefore, in order to sort of understand what's the best property that the Masera example satisfies, we also introduced the global asymptotic stability definition which is just stability and global attractivity. And this is called GAS. Then you have uniform asymptotic stability which is uniform stability along with uniform attractivity. We of course verified that the classical Masera example is not uniformly asymptotically stable. I would rather go to this first that is GUAS which is globally uniformly asymptotically stable. So this should be behind after this because we are looking at increasingly powerful properties. Therefore, I would rather put it above this. So what is globally uniformly asymptotically stable? It is uniformly stable as before but then in the attractivity I add the global property. So you have global uniform attractive stable. So it is uniformly stable and it is globally convergent, globally uniformly convergent. So you can start from any initial condition and you will reach the equilibrium which was assumed to be the origin for all of these. So this is called globally uniformly asymptotically stable, GUAS. So one of the things that I usually point out at this time is that for all of these cases, there is no specified speed of convergence. For example, something like a t0 over t, if it so turns out that your function xt is something like a t0 over t. Then also you have global uniform asymptotic stability. This is not going to be too difficult to verify. You can in fact do that. I mean let me be actually more precise. Say this is t0 over t times x0. Suppose this turns out to be the solution. I am not specifying any differential equation or anything but suppose I tell you this is the solution. So this is also globally uniformly asymptotically stable. Why is that? Stability is obvious because it is sort of decreasing. In fact, let me say t plus 1 just to make our life easy where I say t0 is greater than equal to 0. So for this kind of a system, you can see that as time progresses, I am going to get smaller and smaller value of the states. And therefore as time goes to infinity, of course it is attractive. It does not matter what initial size of the x0 ball was taken. I am guaranteed to convert for all possible initial x0 balls. Okay, therefore this is globally attractive. Uniformity is of course not going to be a problem in this case either. So actually that is something I will have to verify. Let me actually make this. Now let me make this something like this too to be sure that it is also uniform. Suppose the solution is t minus t0. Suppose my solution is this, just so that I can be sure of uniformity too. So it is not difficult to verify that this is also going to be uniformly stable and also globally uniformly attractive. Okay, so for this system, this is globally uniformly asymptotically stable. So this is definitely globally uniformly asymptotically stable. But now if I choose another system which is something like this. This is also globally uniformly asymptotically stable because it is still converging. The global stability property is definitely not lost. And uniformity is also not lost. I can promise you. Any system whose solution has t minus t0 in it is definitely going to be uniform. This is a small tidbit for you. Because if you have seen linear system solutions, they always have t minus t0 as part of their solution. And never t and t0 separately for all time invariant systems. So great. So if you have a solution of either of these kinds, these are both globally uniformly asymptotically stable. However, one thing should be evident to you is that if you look at this guy, this is this denominator is going to infinity much faster than this guy. Why? Because of this square term. It's a quadratic. So this guy is going to infinity much faster than this denominator. And therefore this is going to 0 much faster than this guy. Okay. So this is one of the sort of if you may drawbacks if you may want to call it. Although we may not be concerned with it so much. Of the asymptotic stability definitions. Whenever we say asymptotically stable, we never really talk about how fast we are going to 0. How fast we are going to the equilibrium origin in this case. Yeah. And this is what exponential stability sort of formalizes. Okay. And this is of course again for one particular kind of rate. Yeah. It's not necessarily all kinds of convergence rates. Yeah. Here we are always exponential stability as the term implies always means exponential rate of convergence and same with global exponential stability. So it's exponential rate of convergence. Okay. So it's not exactly codifying all possible rates, but because we are giving nicely upon of conditions later on. So we care of we sort of talk about these additional two definitions which allow us to codify some measure of how fast your solutions are in fact going to the origin. And we are not just saying that the solutions will go to the origin. They'll remain stable and all that. Yeah. We are in fact saying how fast they are going to go to the origin. Okay. So what is exponential stability? It's rather simple. It says that if there exists constants r, a and b which are positive. Right. Now note nothing depends on time and initial time and all that. So exponential stability the way it's defined is naturally uniform. Okay. And hence it's a very strong emotion. Okay. So exponential stability requires the existence of these r, a, b constants such that your state norm always remains within the a norm x0, e minus b, t minus t0. Okay. And this has to hold for all t, t0 greater than equal to 0 and x0 less than r. I would rather make this a little bit more precise and I would say this is actually t greater than equal to t0, greater than equal to 0. Okay. And for all x0 in an r ball. Okay. So for all local that is a bounded x0 in some sense, some x, some ball of radius r and for all initial time greater than equal to 0 and t greater than equal to the initial time, of course. You have this kind of an exponential decay. Yeah. So you can see this right hand side is exponentially decaying because b is a positive quantity and t minus t0 is going to increase as t goes to infinity. So this is an exponential decay to 0, the right hand side. Yeah. And this will happen if your initial condition lie within an r ball. Yeah. And so you have an exponentially decaying envelope or exponentially decaying shrinking ball. Yeah. In which the state trajectories have to lie. And this is called exponential stability. It has been, of course, given the acronym ES. Yeah. So one thing that is sort of well known and it's something you will need to prove you should prove is that exponential stability implies uniform asymptotic stability. Okay. Yeah. Exponential stability implies uniform asymptotic stability. So this is stronger than uniform asymptotic stability. All right. The next one is of course the global version of it. Okay. And what does the global version entail? It just removes the initial condition ball as you would expect. Whenever we make anything global, so one constant actually drops off. So it has only two constants now, right? Only two constants. And it, you have the same condition. In fact, this is exactly the same. And the initial conditions are allowed to lie within any, anywhere in Rn. Okay. Anything where in Rn is allowed for your initial conditions. Okay. So this is the only difference as is always the case. Right. Whenever we are talking about global properties, all we do is we remove the bound on the initial conditions. Right. So this one, of course, has an acronym of G E S global exponential stability. And of course it is true that G E S global exponential stability implies G U A S that is global uniform asymptotic stability. Okay. So again, this is something that you will have to prove. Yeah. Exponential stability and global implies global uniform asymptotic stability. All right. This is a rather nice and strong property if you can indeed conclude this for your nonlinear dynamical system. It's not very easy. It's very unusual. I might add to have exponential stability for a nonlinear system. Okay. For linear systems, this is an obvious thing. Again, for those of you who have seen solutions for linear systems, this is an obvious thing that linear systems always have exponential trajectories either exponentially blowing up or exponentially falling down. Yeah. And so it's not a big deal in those cases. But for nonlinear systems, this is a rather rare property if I may. Okay. All right. So what we want to do is we of course want to see a few interesting examples. Yeah. The first, I mean of systems which satisfy some of these properties or not and so on and so on. Yeah. The first one is this rather complicated looking beast. Yeah. Of course, it should be evident to you that there's no way I can solve this kind. Okay. So we are no longer looking at these very scalar type systems which we can solve and then conclude and then find delta given an epsilon and so on. Because as I had already mentioned, those methods have rather limited applicability. Yeah. Because beyond a point you cannot expect to solve a nonlinear system. Yeah. And therefore we start to look at these more complicated examples. Okay. So let's see what we have. Let's see what we have. So for this complicated system, I'm not even going to attempt to solve it or anything. I will try to show you what the system trajectories look like. Right. So basically the system trajectories look something like this here. This is the x-axis. There is something like a... Okay. So this did not work. Okay. So this is something like a bifurcation here. I'm going to... Make this bigger. Sorry. I apologize. I'm going to try this again. Yeah. Not doing very well with the picture, of course. Let me try to make a trick. Okay. So there is something like a leaf-like bifurcation here and the system trajectories outside of this sort of do this. It's not the complete picture. And the system... So it's still going to zero as you can... I mean, this picture, I hope it's indicating to you that it is going to zero. So zero, zero is the equilibrium. It's not difficult to verify. And the system trajectories inside do something slightly different. I see. Okay. This happened. All right. The system trajectories outside of this sort of do this. All right. The system trajectories inside just form similar small petals. Okay. Just form similar small petals. So these are like the state-space trajectories, right? This is the state-space trajectory. So I'm plotting X1 on this axis and X2 on this axis. Okay. So all trajectories outside do this, curve around and go to the origin. All trajectories inside do this. Okay. So now it's obvious that this system is... Even just by looking... Just as we did this for the Van der Poel oscillator, we can do it for this sort of a system also, right? So it should be obvious to you that the system is attractive, right? This in blue. But it is not stable. Okay. Why is it not stable? Right. So I mean, again, you can... Suppose I do this by making some epsilon ball. I make an arbitrary epsilon ball around the equilibrium, which is origin in this case. So let me make an arbitrary epsilon ball. Okay. So I made this epsilon ball, as you can see. I'm trying to center it, of course. I think it was more centered earlier than it's now. Yeah. I think this is centered enough. All right. With the drawing skill that I have. Right. All right. So excellent. So this is sort of centered at the origin. This is... Like I said, this is an epsilon ball. Right. Right. Now, what is the problem? I think all of you can see the problem. You can all see the problem from the trajectories I've drawn. It doesn't matter where I start. Suppose I start outside. I will get out like this from this petal and get in here. Okay. So suppose I start inside the petal, then I still go out and then come back. So this is certainly not stable. Okay. Now, remember, again and again, I go back to the stability definition because usually there is a lot of confusion on that. Yeah. Remember, I have to be able to find a delta for every epsilon the user gives me. Okay. A lot of students would come to me and say, but sir, I can always, you know, draw this epsilon ball. Yeah. This is very, very large epsilon ball. Yeah. And if I draw this large epsilon ball, sir, I can always, you know, get all these trajectories to, you know, converge to the or remain inside the epsilon ball. Yeah. Or you can find definitely some deltas which will sort of remain inside the epsilon ball. Yeah. But that's not enough. Okay. You need to be able to find me a delta ball for every epsilon ball I give you. Okay. So, this yellow thing that is out here is not the epsilon ball that I would give you. I would give you this really small, tiny green epsilon ball. Okay. And then, of course, you will have some trouble. Okay. You will not be able to find me a delta because everything that you give me, everything that starts inside this epsilon ball, forget a smaller delta ball. Everything that starts inside the epsilon ball will definitely exit the epsilon ball one way or another. So, although we have the rather desirable property of attractivity, that is, you have nice convergence, but what's happening is that the system is going really far out to come back and converge. Yeah. Which is not allowed in stability. Yeah. You cannot go really far out and then come back. Then the system is not stable anymore. Sure. It's attractive. If that's all the property you care about, then well and good, but it is not a stable system. Okay. And in a lot of circumstances, stability is a key property. Yeah. Stability is a key property. If I have an epsilon ball and I want to remain in it, I should be able to find corresponding initial condition balls. Yeah. Otherwise, I don't have... So, stability actually gives you a... For those of you who have experience of this frequency domain sort of terminology, the stability property actually tells you something about the transient behavior, right? Because it's saying that if I'm given an epsilon ball, then I can always find a delta ball so that my trajectories always remain within the epsilon ball, always. So, that's telling me something about the transient behavior, the bounds on the transient behavior, all right? And the exponential stability or the conversion, the attractivity property that we have, that we've spoken about is telling me something about the steady-state behavior. Okay. And all of you know that even in linear systems theory, it's rather critical to have a handle on the transient behavior and the steady-state behavior both. It's not enough to have one or the other. In fact, a lot of people are happy to compromise on the steady-state behavior. They can deal with a little bit of accuracy in the steady-state behavior. That is, even if your trajectories don't exactly go to the origin or the equilibrium, they're okay. But they definitely care a lot about the transient behavior. You cannot have transients which are simply shooting up to, you know, dangerous values or absolutely bad big values and then coming back to the origin as T goes to infinity, all right? So, in fact, most practitioners would reject any controller which makes your system jump to large values before coming to the origin. And they would basically stop the controller right when it starts to jump. Okay. So, this is absolutely unacceptable. So, both stability and attractivity together are key. And therefore, the reason for us defining all these, you know, new properties here. Okay. Excellent. So, this system is attractive and not stable. Okay. Let's keep that in mind. And that's what it says here. All right. Let's look at quickly the dynamics of a pendulum. And we've already seen the pendulum. You've physically seen pendulum, right? I mean, like, you know, even the grandfather clock that you typically have at your home, although that one is never stopping, right? But if you remove the battery, right? Then it does this, whatever. I mean, it oscillates for a bit. Then it settles down. Okay. This is the standard pendulum. So, that is the pendulum with damper. That is, and the dynamics of that with, of course, normalized mass and things like that is something like x2 dot is x2 dot is minus sin x1 minus kx2. Okay. This, the phase plane portrait of this system, again, this is not easy to solve. You know, you cannot easily get solutions for a simple system such as this either, right? But what we will, what we will simply try to draw some phase plane portraits. So, what you have, this is again between x1 and x2. And your trajectories, wherever you start, they look like spirals that are falling in. Yeah, they look like spirals that are falling in. So, this is, I mean, it goes on for infinite time. And therefore, I mean, this is both attractive. This is attractive. In fact, uniformly attractive, uniformly stable. And in fact, globally uniformly attractive and uniformly stable. It doesn't matter where I start. I'll just start with the big spiral, but I will still go to the origin. And so, this is in fact globally uniformly asymptotically stable. So, what I say here is not complete. This is in fact, globally uniformly asymptotically stable system, right? Anyway, if you remember, we had sort of spoken of such cases when time doesn't appear in the vector field on the right, you get uniformity for free. So, it's globally uniformly asymptotically stable system. All right, great, great. So, what we saw today was the rest of the stability properties, whatever was remaining. We also saw that in order to specify a rate of convergence to the equilibrium, we also can define exponential stability and global exponential stability properties which are very strong, right? And we worked out a few examples, right? We saw, you know, a rather interesting example of a stable and attractive but non-stable system. And then we saw the standard pendulum example which is both attractive and stable. So, in fact, globally uniformly asymptotically stable, right? And so, we have we also sort of learned that both transient and steady-state behavior are critical and transient behavior translates to stability while steady-state behavior translates to activity properties. All right? Excellent. So, we will do some more of this in the next session. Thank you.